Let be an elliptic differential operator, where and are vector bundles over a closed oriented manifold . Suppose and have smooth inner product structures. Let and be the disk and sphere bundle of the cotangent bundle, respectively. Let be the projection. Let be the associated symbol class. Let be its Chern character. Let denote the pullback of the Todd class . Let be the fundamental class.

Definition. The topological index of is defined to be .

Atiyah-Singer Index Theorem. , where is the analytical index.

Example. [Point Case] Let be a point. Then and are finite dimensional vector spaces. Any non-trivial differential operator is a linear map between them, and hence of order 0. Thus, . Note that is empty set. Recall the definition of Chern character, then we have and . Since is empty set, then and hence . Recall the definition of Todd class, we then have . Therefore, by the definition of topological index, we have . By Atiyah-Singer, .

Example[ Case] Let , , Then is a first order elliptic operator. We now claim that . We give four different proofs.Proof 1. We compute directly. Note that and that via .Proof 2. Since , we have Proof 3., since is self-adjoint.Proof 4. We will show that the topological index vanishes whenever is odd. See next example.

Example[Odd Dimensional Case, Theorem 13.12 in Lawson-Michaelson]
We will show that the topological index of any elliptic differential operator vanishes whenever is odd.

We want to show that , where is an elliptic differential operator of order . Consider the diffeomorphic involution given by . Since

,
it suffices to show . In fact, , since is homotopic to via .
Next, we need to introduce the Thom Isomorphism to talk about the de Rham operator.

Let be an oriented -vector bundle over , with inner product on each fiber. We now give the notion of Thom class and Thom space.

Definition. is a Thom Class of if it restricts to a generator of on each fiber. The quotient is called the Thom space of , and denoted by .

Thom Isomorphism Theorem. The composition is an isomorphism.

We denote the composition by and denote its inverse by .

Remark. has the following two other interpretations.

(1). Integration over the fiber., where is a bundle over .
Let be given and choose and . Associated to these data is a form , defined as follows. Given and a basis , choose lifts such that , for each , and define

Now is defined by

Integrating over the fibers will give the second formulation of the topological index, which is the next theorem. The factor compensates for the difference between the orientation on induced by the one on , and the canonical orientation on inherited from its almost complex structure.

(2) . Also, we can use the second interpretation of to give that formulation.

is the composition (Poincare duality) (Poincare-Lefschetz duality),

Then, we compute.

Theorem..

We will then give the third formulation of the topological index. To do this we need the notion of Euler class.

Definition. The Euler class of an oriented -bundle over , denoted by $latexe(E)$, is the image of the Thom class under the following isomorphism: . We may denote the composition by .

Theorem.[Gysin Sequence] To any bundle as above there is associated an exact sequence of the form

Definition. The Euler characteristic of is defined to be

From now on, we assume .

We want to analyze to give the third formulation of the topological index. For details please see Lawson-Michaelson, P258, Theorem 13.13.

Since – and are inverse to each other, we have . Applying to both sides, we then get Thus we can write , if .

Theorem. , if is not zero.

Now, we are trying to apply this formula to the de Rham operator.

Example.[de Rham operator] Let Then we already know that is an elliptic complex, that is an elliptic operator and that . We want to use the above theorem to show that the Euler charateristic.

For a complex vector bundle , by the splitting principle, we can write as . Then . It follows that Similarly, we have

Back to our example, applying the real splitting principle to , we compute

since for . Note that , and that , then by the theorem above, we obtain

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We begin by endowing a vector bundle with a Clifford module structure. It is with additional structure that we may define a Dirac operator.

Let be an -dimensional Riemannian manifold with covariant derivative (on ) , and let be a vector bundle.

Clifford Module Bundles and a Dirac “Type” Operator

Definition (Clifford module)

A Clifford module for a real inner product space is a left module over . Equivalently, there is a -algebra homomorphism given by . Since for any (see the Glossary below), one has that satisfies .

Definition (Bundle of Clifford Modules)

A bundle (as above) is a bundle of Clifford modules if there is a map of bundles of -algebras such that for any section . In other words, for each , is a Clifford module for .

Definition (Dirac type operator)

Let be a Clifford module bundle equipped with a covariant derivative . Let be the map defined by the composition

where is the inverse of the bundle isomorphism , and where is Clifford multiplication. We call such a map (which depends on the Clifford module bundle , and ) a Dirac type operator.

If we a fix an orthonormal frame for over some neighborhood and, using the metric , let be the corresponding frame for , we may write this composition locally as

That is, a Dirac type operator is locally of the form

.

Proposition

A Dirac type operator is a first order differential operator.

Proof

Let and . Using the local description above, we compute:

.

In particular, for , so is -linear and hence in . Thus , and as itself is not -linear, is of order 1.

Remark

The above proof extends (by incorporating induction) to show that the composition of a – and an order differential operator is a differential operator of order . Here, and are differential operators of order 1 and 0, respectively.

We next show that is elliptic.

Lemma (Symbol of a Dirac type operator)

Let be a Dirac type operator and let . Then the symbol of at is given by

,

where is the dual to determined by the metric, i.e., such that .

Proof

Fix , and let be an open neighborhood of in . Using choose an orthonormal frame of with dual frame for . Being a bundle homomorphism (over ), is -linear in the -ordinate. Thus it suffices to verify the proposition for ; that is, we wish to show that for .

Choose a local chart such that ; let . Note that is an orthonormal basis for . Let ; then , so .

Thus (see the Glossary below), as is order , we have

,

as required.

Corollary

A Dirac type operator is elliptic.

Proof

If then has inverse .

Remark

Noting that (the product of linear maps), we observe that . Taking , this — apparently — implies that a Dirac type operator is, at the symbol level, the square root of the Laplacian.

-grading and “a” Dirac Operator

A Dirac type operator is formally self-adjoint (so that its index is ) if we impose the following further restrictions on the Clifford-module bundle.

Definition (Clifford-Compatible)

Let be a bundle of Clifford modules. We say that is Clifford compatible if it is equipped with a metric and a covariant derivative such that

(1) is Riemannian, i.e., for all sections :

, and

(2) for all vector fields and for any section :

.

Definition (A Dirac operator)

A differential operator is a Dirac operator if

(1) is a Dirac type operator, and

(2) is Clifford compatible.

Lemma

If is oriented and is a Dirac operator, then is formally self-adjoint. That is,,
where (and the integration is with respect to the volume form on ).

Proof

Omitted 😦

Consequently we have (see Ning’s blog). To make use of the index, then, we introduce a -grading on Clifford module bundles.

Definition

Recall that a Clifford algebra is -filtered — with — and -graded — (even and odd products).

A Clifford module bundle is -graded if it decomposes into a direct sum of vector bundles such that for each and , one has .

Such a -graded bundle is compatible if this decomposition is both orthogonal with respect to and parallel with respect to the covariant derivative , ie. .

Example

If is oriented, the Clifford bundle is a -graded compatible Clifford bundle. Some words which may be connected to verify this: Levi Civita connection, induced connection on , lift to principal spin bundle, induced covariant derivative on associated vector bundle, compatibility with the metric.

Examples of Dirac Operators

We now look at four examples of Dirac operators. The first two are familiar; here we reinterpret them in terms of Clifford modules.

(1) and (2). Let is an orthonormal basis for . Since (by an equivalent definition of the exterior algebra) , we see that is induced by the map taking to and descending to ; that is, . It follows that is an isomorphism and preserves the grading.

(3) Using that , I feel like we need to be working with here. Please comment!

Corollary

The exterior algebra bundle over a(n oriented?) manifold and the clifford bundles are isomorphic as vector bundles.

Proof

Let denote the principal -bundle associated to . By parts (1) and (3) of the lemma, the map above induces a vector bundle isomorphism .

Theorem 2.5.12 (Lawson, Michelsohn)

Under this bundle isomorphism , the de Rham operator corresponds to the Dirac operator .

Corollary

Since we have already established (see Hailiang’s(?) blog post) that the Euler characteristic of is equal to , the theorem (along with the grading-preserving property of ) implies we may also compute it as .

Example 2: The Signature Operator

We now look to reinterpret the signature operator in terms of Clifford bundles.

Recall (see Hailiang’s blog) in the case that and is even, the Hodge star operator is an involution so we can decompose into the and eigenspaces of . We defined the signature operator . Since , we saw that took to and letting , we found that . The signature of was defined to be the signature of the quadratic form on given by .

In the case that is odd, we had to modify the construction. We complexified, taking , and defined . Then the above paragraph went through with replacing and replacing .

Let be an oriented orthonormal basis for . Let in . Then by the lemma above and the corresponding properties of the volume form in , we obtain that is a basis-independent section of .

Lemma

(1) We have

(2) If is even and , then .

Proof

(1) We compute . Writing for , one finds that is even if and only if or .

(2) It suffices to verify for . We have and . (Here, a hat indicates that the element be omitted from the product.) Since is even, .

Now, acts on any Clifford module via , and by part (1) of the lemma this defines an involution in the case ; in the case , defines an involution. Compare with the Hodge star operator recalled above. So define

then . Thus if is an oriented manifold and is a Clifford module bundle of , putting , we have

if : , or

if :

Corollary

If is even then is -graded, i.e., for and , one has .

Proof

By part (2) of the lemma, for we have .
Thus if (i.e. ) then , so .

Proposition

If is even and is -graded compatible, then the associated Dirac operator splits as.
In particular, if , by Theorem 2.5.12 we have .

Proof

Since is -graded and is compatible (so in particular, the covariant derivative preserves , the Dirac operator
takes to .

Example 3: twisted Dirac Operators

Preliminary: If and are vector bundles over with covariant derivatives and , respectively, then the tensor product bundle has covariant derivative.

Fact: If is a compatible -graded Clifford module bundle and is a Riemannian bundle (see Property (1) of a compatible Clifford Bundle above for the definition), then is a compatible -graded Clifford module bundle (with Clifford multiplication for , , ). In the case that , we call the Dirac operator on a twisted Dirac operator.

Fact: (Apparently from topological K-theory) If the Index theorem holds for any twisted Signature operator then it holds for all elliptic differential operators.

Example 4: Spin Manifolds and The Atiyah-Singer Dirac Operator

Recall (see Prasit’s blog) that there is an isomorphism . So since is an module, one has that is an -module via , for and . To avoid confusion, let us call with this module structure .

Now, any -module is isomorphic to , so it follows that any -module is isomorphic to .

Let be a Clifford module bundle. From the above paragraph we see that each fiber (a -module) is isomorphic (via , say) to , where is a copy of . We may then
ask if this splitting extends over the whole bundle; that is, is there a Clifford module bundle and a bundle isomorphism which restricts fiberwise to an isomorphism .

In turns out the answer is a resounding “Yes” if is a spin manifold.

Definition (Spin Structure)

Let is an -dimensional vector bundle. A spin structure on is a principal -bundle together with a bundle isomorphism . (Then is the associated vector bundle for ). Using classifying space theory, we may reinterpret this
to say that a spin structure on is a lift of the classifying map to .

We may break up the existence of a spin structure into pieces as follows.

After choosing a metric on , we may first reduce the structure group of to . (The only obstruction to doing so is the paracompactness of .) So we’re left to lift a map to .

Since is the universal cover of , we may first try to lift to .

The short exact sequence of groups induces a fibration of classifying spaces.

It turns out that the map lifts to if and only if the composite is nullhomotopic. Since , there is an element that vanishes if and only if lifts. We call the first Stiefel Whitney class of .

Similarly, the map lifts to if and only if the composition is nullhomotopic; we call the corresponding element in (that vanishes iff lifts) the second Stiefel Whitney class of .

Definition (Spin manifold)

We will call an oriented manifold (so ) a spin manifold if its tangent bundle admits a spin structure (i.e., ). It can be shown that this is equivalent to the existence of a trivialization of over the -skeleton of . (Compare with the fact that is orientable if and only if is trivializable over the -skeleton.)

Definition (The Atiyah-Singer Dirac Operator)

Suppose has a spin structure with principal -bundle (associated to ). Since acts on on the left and (where has general element with , ) is a subgroup of the group of units , we may define
the -bundle associated to by . Since is a module, is a Clifford module bundle.

Some words: By lifting the Levi-Civita connection on one obtains a connection on , and hence (see who’s blog?) a covariant derivative on which makes it Clifford compatible
as a graded Clifford module bundle.

We may then define the Atiyah-Singer Dirac Operator by

.

Some more words:

If then is called a harmonic spinor.

If has positive scalar curvature, then is injective. So if we have ways to compute the index (using the ASHI theorem, for example), we may be able to deduce that does not admit a metric of positive scalar curvature.

Glossary

(to include links to other blog posts)

Differential Operator (global definition)

If and are vector bundles over (of the same dimension), we define the family of differential operators of order from to by

with . In particular, .

Symbol of a differential operator

(cf. Juanita’s blog) We recall the definition of the symbol of an order- differential operator . Denote , , so . Let . Let . The symbol of at is the homomorphism defined by

where , and is such that .

Covariant Derivative

A covariant derivative on a vector bundle is a map , where is an -linear map satisfying the Leibnitz rule for and .

Graded Algebras

(cf Prasit’s blog) A -algebra is -graded if such that . A -algebra is filtered over
if such that and . A graded algebra defines a filtered algebra by taking , and conversely a filtered algebra defines a graded algebra by taking (with ).

Tensor Algebra and the Clifford Algebra

If is a vector space, the tensor algebra has multiplication defined by concatenation, i.e., . Thus is a graded algebra with , and filtered with .

Recall that with . (Here, is a quadratic form; sometimes it is convenient to refer instead to the associated symmetric bilinear form .) For any , so maps to under the quotient . This sets up a natural identification between and . Whence the associated graded algebra for is isomorphic to . In particular, they are isomorphic as vector spaces, with dimension .

Clifford bundle

The Clifford bundle has fibers (), where is a Riemannian metric on . (The latter isomorphism is given by identifying an orthonormal (with respect to ) basis for with the standard generators .) Just as is the -bundle associated to the orthogonal frame bundle (of the tangent bundle) over , is the associated -bundle to .

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As a provisional definition, clifford algebraover a field can be defined as

where,

Easy to see that has dimension . can also be thought of as

where as vector space, is the tensor algebra, is same as above, except that ‘s are standard basis for . This observation leads to a more general definition of clifford algebra, where is a vector space equipped with a symmetric bilinear form

Definition 1 Let be a vector space with symmetric biliear form and quadratic form . Then the clifford algebraover can be defined as

Remark 1These are some of the properties that enjoys

There is a natural inclusion of .

If then the exterior algebra

Let denote the clifford multiplication( induced by the tensor product of ), then

Universal Property : Let , where is a -algebra, such that , then there exists an unique map such that , that is the following diagram commutes

A map , such that , extends to a -algebra homomorphismThus the orthogonal group has an action on

Proposition 2 is a filtered algebra whose associate graded is

Before proving the theorem, recall the following definition

Definition 3If is a -algebra then a filtration of is sequence of subspaces

The associate gradedof is defined as

Proof: Let be the quotient map

Define, ( fold tensor product). Define filtration on by setting

Define filtration be the filtration on the clifford algebra. Note . Hence, in the associated graded . On the other hand the relation prevails in the associated graded. Hence the associate graded is isomorphic to .

Remark 2 is a -graded algebra.

Definition 4Recall,. Define,

and

On we have an involution map, which is induced by the involution on given by,

If then

Let and , then observe

Lemma 5There exist short exact sequences

and

where is the map which sends

Let be a field. Recall, tensor product of -algebras and is a -algebra, denoted by and multiplication is given by

moreover if denotes the set of all matrices. Then we have the following isomorphism

Define

Remark 3If , then

This follows from the fact that the quadratic forms and induces isomorphic innerproduct structure on where the isomorphism sends

Theorem 6If , then we have the following isomorphisms

Corollary 7(Bott Periodicity) As a consequence of (iv) we have

if even, and

if is odd.

To work out the case when the underlying field is . For any field we have the following isomorphisms.

Lemma 8For any field

Proof: Let denote the standard basis of and cannonical generatoring set of the Clifford algebra.

To get the first isomorphism we simply produce a map given by sendingandIt is easy to check that the above map is an isomorphism.

is similar to .

One can explicitly check some of the lower dimension cases( ). Then one can repeatedly use the isomorphisms in previous lemma. One has to work upto dimension when , before one sees the patern, which is called the Bott periodicity. TSome of the calculations are as follows calculations are as follows

In general one gets,

Putting all these observations together we get

Theorem 9 The Bott periodicity in case of real number looks like

Lemma 10 As -algebras Proof:The isomorphism is given explicitly by the map induced by sending